This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
from bisect import bisect from sympy import divisor_count def A365485(n): m = 10**n plist, qlist, c = tuple(divisor_count(i) for i in range(1,m+1)), [0]*(m+1), 0 for i in range(m): qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i c = max(c,a) return c
from bisect import bisect from sympy import divisor_sigma def A365398(n): plist, qlist, c = tuple(divisor_sigma(i) for i in range(1,n+1)), [0]*(n+1), 0 for i in range(n): qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i c = max(c,a) return c # Chai Wah Wu, Sep 08 2023
a(6) = 5 because phi is nondecreasing on 1,2,3,4,5 or 1,2,3,4,6 but not on 1,2,3,4,5,6.
# Computes the first N terms of the sequence. function A365339List(N) phi = [i for i in 1:N + 1] for i in 2:N + 1 if phi[i] == i for j in i:i:N + 1 phi[j] -= div(phi[j], i) end end end lst = zeros(Int64, N) dyn = zeros(Int64, N) for n in 1:N p = phi[n] nxt = dyn[p] + 1 while p <= N && dyn[p] < nxt dyn[p] = nxt p += 1 end lst[n] = dyn[n] end return lst end println(A365339List(69)) # Peter Luschny, Sep 02 2023
Table[Length[LongestOrderedSequence[Table[EulerPhi[i],{i,n}]]], {n,100}]
import math
def phi(n):
result = n
for i in range(2, math.isqrt(n) + 1):
if n % i == 0:
while n % i == 0:
n //= i
result -= result // i
if n > 1:
result -= result // n
return result
# This code uses dynamic programming to print the first N=100 values of M.
N=100
M = [0 for i in range(N)]
dynamic = [0 for i in range(N+1)]
for n in range(1,N+1):
i = phi(n)
new = dynamic[i] + 1
while (i<=N and dynamic[i] < new):
dynamic[i] = new
i+= 1
M[n-1] = dynamic[N]
print(M)
from bisect import bisect from sympy import totient def A365339(n): plist, qlist, c = tuple(totient(i) for i in range(1,n+1)), [0]*(n+1), 0 for i in range(n): qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i c = max(c,a) return c # Chai Wah Wu, Sep 03 2023
from bisect import bisect from sympy import totient def A365737(n): plist, qlist, c = tuple(-totient(i) for i in range(1,n+1)), [0]*(n+1), 0 for i in range(n): qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i c = max(c,a) return c
from bisect import bisect from sympy import totient, isprime def A365740(n): plist = tuple(totient(i) for i in range(1,n+1) if not isprime(i)) m = len(plist) qlist, c = [0]*(m+1), 0 for i in range(m): qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i c = max(c,a) return c
from bisect import bisect from sympy import totient def A365741(n): k = 10**n plist = tuple(totient(i) for i in range(1,k+1) if not isprime(i)) m = len(plist) qlist, c = [0]*(m+1), 0 for i in range(m): qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i c = max(c,a) return c
from bisect import bisect from sympy import totient def A365738(n): k = 10**n plist, qlist, c = tuple(-totient(i) for i in range(1,k+1)), [0]*(k+1), 0 for i in range(k): qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i c = max(c,a) return c
from collections import Counter from sympy import totient def A365742(n): return max(Counter(totient(i) for i in range(1,n+1)).values())
from collections import Counter from sympy import totient def A365748(n): return max(Counter(totient(i) for i in range(1,10**n+1)).values())
a(7) = 6 because A001615 is nondecreasing on 1,2,3,4,5,6 or 1,2,3,4,5,7 but not on 1,2,3,4,5,6,7.
Length[LongestOrderedSequence[#]] & /@ Rest[FoldList[Append, {}, Table[n DivisorSum[n, MoebiusMu[#]^2/# &], {n, 20}]]] (* Eric W. Weisstein, Mar 09 2025 *)
from math import prod from bisect import bisect from sympy import primefactors def A371156(n): def f(n): r = primefactors(n) return n*prod(p+1 for p in r)//prod(r) plist, qlist, c = tuple(f(i) for i in range(1,n+1)), [0]*(n+1), 0 for i in range(n): qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i c = max(c,a) return c
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